
where \(\gamma \)  is the
modulus of the gyromagnetic ratio, \(m^{(k)}_{x}(x)\)  and \(\tilde{m}_{l,\alpha }^{\rm K}({\bf r})\)  represent the amplitude of the spin-waves and Kittel modes, and \(\hat{a}_{k}\)  and \(\hat{b}_{l}\)  denote the magnon operators in the film and wire. For simplicity, \(k\)  denotes \(k_y\) .
By the dipolar field of magnetic charge [1]}, [2]},
the total Hamiltonian
\(\hat{H}/\hbar &=\sum _l\omega _{{\rm K}}\hat{b}_{l}^{\dagger }\hat{b}_{l}+\sum _{k}\omega _{k}\hat{a}^{\dagger }_{k}\hat{a}_{k}\nonumber \\&+\sum _l\sum _{k}\left(g_{k}e^{-ikR_l}\hat{b}_{l}\hat{a}^{\dagger }_{k}+g_{k}e^{ikR_l}\hat{b}_{l}^{\dagger }\hat{a}_{k}\right)\) 
